An explicit decomposition of higher Deligne-Lsuztig representations

2025/06/15 by Liu, Ben, Nie, Sian
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2506.12918

Abstract

In a previous paper, the second named author obtains a decomposition of an elliptic higher Deligne-Lusztig representation into irreducible summands, which are built in the same way as Yu types using a geometric analog κ' of the Weil-Heisenberg representation κ. In this note, we show that κ' and κ differs by a character χ. Moreover, under a mild condition on the cardinality q of the residue field (for instance q > 3), we show that χ equals the quadratic character constructed by Fintzen-Kaletha-Spice, which gives an explicit irreducible decomposition result on elliptic higher Deligne-Lusztig representations. As an application, we deduce (under the mild condition on q) that each unramified Yu type appears in the cohomology of higher Deligne-Lusztig varieties, and each unramified Kaletha's regular supercuspidal representation is the compact induction of a specified higher Deligne-Lusztig representation up to a sign.

Citations

• Green functions for positive-depth Deligne--Lusztig induction
• Convex elements and cohomology of deep level Deligne-Lusztig varieties
• Decomposition of higher Deligne-Lusztig representations
• The scalar product formula for parahoric Deligne--Lusztig induction
• The algebraisation of higher level Deligne--Lusztig representations II: odd levels
• A twisted Yu construction, Harish-Chandra characters, and endoscopy
• Geometric L-packets of Howe-unramified toral supercuspidal representations
• Cohomological representations of parahoric subgroups
• Types for tame p-adic groups
• The cohomology of semi-infinite Deligne--Lusztig varieties
• The algebraisation of higher Deligne--Lusztig representations
• Regular supercuspidal representations
• Maximal varieties and the local Langlands correspondence for GL(n)
• Supercuspidal representations: an exhaustion theorem

Related

• Decomposition of higher Deligne-Lusztig representations
• The algebraisation of higher level Deligne--Lusztig representations II: odd levels
• Convex elements and cohomology of deep level Deligne-Lusztig varieties
• The algebraisation of higher Deligne--Lusztig representations
• Deligne-Lusztig Constructions for Division Algebras and the Local Langlands Correspondence